COVID-19 Forecasts for Cuba Using Logistic Regression and Gompertz...

MEDICC Review, July 2020, Vol 22, No 332

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COVID-19 Forecasts for Cuba Using Logistic Regression and Gompertz CurvesJuan Felipe Medina-Mendieta MS, Manuel Cortés-Cortés PhD, Manuel Cortés-Iglesias MS

ABSTRACTINTRODUCTION On March 11, 2020, WHO declared COVID-19 a pandemic and called on governments to impose drastic measures to fi ght it. It is vitally important for government health authorities and leaders to have reliable estimates of infected cases and deaths in order to apply the necessary measures with the resources at their disposal.

OBJECTIVE Test the validity of the logistic regression and Gompertz curve to forecast peaks of confi rmed cases and deaths in Cuba, as well as total number of cases.

METHODS An inferential, predictive study was conducted using lo-gistic and Gompertz growth curves, adjusted with the least squares method and informatics tools for analysis and prediction of growth in

COVID-19 cases and deaths. Italy and Spain—countries that have passed the initial peak of infection rates—were studied, and it was inferred from the results of these countries that their models were ap-plicable to Cuba. This hypothesis was tested by applying goodness-of-fi t and signifi cance tests on its parameters.

RESULTS Both models showed good fi t, low mean square errors, and all parameters were highly signifi cant.

CONCLUSIONS The validity of models was confi rmed based on logis-tic regression and the Gompertz curve to forecast the dates of peak infections and deaths, as well as total number of cases in Cuba.

KEYWORDS COVID-19, SARS-CoV-2, logistic models, pandemic, mortality, Cuba

INTRODUCTIONThe COVID-19 pandemic and the characteristics of the SARS-Cov-2 viral agent[1] have led many governments to restrict social contact in order to cut the chain of transmission and thus reduce cases and deaths. The measures include some variation of lockdown, which in various countries has proven effective at curbing disease spread, fl attening the curve and avoiding health system saturation.[2] Thus, it is vitally important for decision-makers to be able to approximate the maximum number of infections and deaths expected, as well as when caseload peaks will occur. In Cuba, many measures have been implemented to mitigate COVID-19 spread and to limit the severity of cases and deaths.[3] However, until April 22, 2020, the increase in confi rmed case numbers was approximately exponential. Going forward, reliable estimates are needed to inform decision-making in the context of limited resources.

Many such forecasts are made using mathematical modeling. A classic epidemiological model is SIR (Susceptible, Infectious, Recovered), based on ordinary differential equations. This modeling has been used successfully for the COVID-19 pandemic in some regions.[4,5] In Cuba, several authors have also applied it to the COVID-19 pandemic.[6–8]

Other techniques that have been used for modeling COVID-19 are:

• Statistical time-series models to predict the number of infec-tions and/or deaths[9]

• Data processing to obtain forecasting models using the inter-net[10]

• Models based on artifi cial intelligence and machine learn-ing[11,12]

These approaches are based on parameters that describe different characteristics of the pandemic. The estimation of these guiding parameters is complex, requiring controlled study of samples or use of approximations. Interpreting the models themselves is also complex.

Among the statistical models are logistic population growth models and the Gompertz growth model.[13] These models have been used in the COVID-19 pandemic and are less complex than those previously mentioned. But they are limited to short-term forecasts since they incorporate few parameters related to changes in epidemic dynamics, such as those that are sensitive to actions of a clinical nature, or to transmission-mitigation measures. To estimate the parameters of these models, the nonlinear least squares method is used. This modeling has been applied worldwide to forecast for incidence and prevalence rates.

Various studies have used logistic models to make predictions regarding COVID-19’s epidemiologic dynamics and the disease’s effects. Batista used the logistic regression model to study the magnitude of the pandemic in China through February 25, 2020;[14] Morais used it in forecasting deaths in China, Iran, Italy, South Korea and Spain;[15] Tátrai and Várallyay applied the model to predict the peaks in various countries affected by COVID-19 and assessed the quality of its fi t with data from various regions in China affected by COVID-19.[16] Wu used a logistic model to estimate the peak in confi rmed cases for Europe and the United States, and evaluated goodness-of-fi t using a sample of 29 provinces in China and 19 countries that had passed the peak.[17] Qaedan used a logarithmic-logistic model to obtain predictions for the state of Utah in the United States and assessed its fi t based on adjustments made in South Korea and Italy.[18]

IMPORTANCEThis study confi rms the validity of an analytical tool that can be used to forecast the total number of COVID-19 cases and deaths in Cuba at different points in time, al-lowing health authorities and government to make more informed decisions on use of available resources to stem the pandemic and organize recovery phases.

33MEDICC Review, July 2020, Vol 22, No 3

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Some studies have implemented the Gompertz model. Mazurek and Nenickova applied it to predict the pandemic’s peaks in the United States.[19] Mazurek took a similar approach to study data for the United Kingdom, the Russian Federation, Turkey and the world as a whole;[20] and Razzak applied the model to predict the course of the pandemic in New Zealand.[21]

Other studies have used both models simultaneously to obtain forecasts for COVID-19. Jia used Gompertz, Bertalanffy, and logistic models to predict COVID-19 case numbers in various regions in China. These authors fi rst studied the models’ goodness-of-fi t using data from SARS-CoV-1 confi rmed cases in China in 2003.[22] Similarly, based on the goodness-of-fi t of the logistic model and the Gompertz model for the data from China and South Korea, Villalobos presented predictions for Costa Rica.[23] Milhinhos and Costa adjusted logarithmic-logistic models and logarithmic-Gaussian models to obtain forecasts for Portugal based on their goodness-of-fi t for distribution of COVID-19 data in South Korea.[24]

Dattoli used a three-parameter logistic model and the Gompertz model to make estimates for Italy.[25] Bauckhage used the logistic and Gompertz models to obtain predictions for Germany for mid-April 2020,[26] while Rodrigues-Silva used these models to obtain predictions for the state of Goias in Brazil[27] and Dutra used them to estimate the number of persons affected by COVID-19 for various US states and the whole country.[28] Attanyake fi tted logistic, Gompertz and other exponential models to data corresponding to the impact of COVID-19 in Sri Lanka, Italy and Hubei, a province in central China.[29] Ahmadi adjusted the Gompertz, Bertalanffy and cubic polynomial models to forecast pandemic dynamics for April 2020 in Iran.[30]

The ordinary differential equations presented in Equation 1 and Equation 2 are known as the logistic differential equation (or Verhulst equation) and Gompertz equation, respectively.[31]

Equation 1: Logistic differential equation

Equation 2: Gompertz differential equation

Both describe the growth of populations where: P(t) represents the number of organisms or the size of a population at a given moment in time, r represents the instantaneous rate of increase and K corresponds to the carrying capacity of the environment or the maximum number of individuals that the population can sustain. K and r are positive real numbers and the function P(t) is positive, monotonically increasing and suitable for representing epidemiological models, as it presents a rapid initial growth that is approximately exponential and as the number of infections increases, the number of non-infected individuals in the population decreases. As a result, the relative growth rate within the population decreases until growth stops when there are no individuals left to infect.

Both models present an explicit solution provided by Equations 3 and 4 for the logistic model and Gompertz models, respectively.

Equation 3: Logistic model (b >0)

Equation 4: Gompertz model (b > 0)

P0 represents the population (P) at the start of the growth process (0 < P0 < K). The b parameter is found to be associated with displacement on the abscissa axis for both sigmoid models. This is obtained through changes in variables (in Equation 3, algebraic transformations were applied before implementing the variable change).

The infl ection point for these population growth models is of interest, as it represents the moment at which the rate of growth is highest, which can be interpreted as the peak of the pandemic. The infl ection point for the logistic model is presented in Equation 5 while the infl ection point for the Gompertz curve is presented in Equation 6. In the logistic model, this point is at 50% of population growth (the logistic function is symmetrical with regard to this point) while this point on the Gompertz model is approximately located between 35% and 40% of population growth.[31]

Equation 5: Infl ection point of the logistic model

Equation 6: Infl ection point of the Gompertz model

The relative rate of population growth is linear in the logistic process (Equation 7) and logarithmic in the Gompertz process (Equation 8). The latter growth process develops more slowly with respect to the logistic model process.[31]

Equation 7: Relative population growth rate in the logistic model

Equation 8: Relative population growth rate in the Gompertz model


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This study aims to fi t logistic and Gompertz models to the distribution of COVID-19 in Cuba for confi rmed and deceased cases, to demonstrate the fi t of these models for these distributions in such a way that they can be generalized as predictive models and to make forecasts for the peak dates of confi rmed cases and deaths due to COVID-19 in Cuba.

The fi rst aspect studied was the fi t of the models used for the distribution of COVID-19 confi rmed cases and deaths in Spain and Italy, countries that had passed the peak of the pandemic. The good fi t of these models in those countries and their comparative simplicity in relation to other models has piqued interest in applying them to forecasting in Cuba. The adequacy of the models in estimating distribution of confi rmed cases and deaths in Cuba was assessed by analyzing the parameters for goodness of fi t and testing the models themselves for statistical signifi cance.

METHODSDesign and participants This is an inferential and predictive study using the logistic model and the Gompertz growth curve. The curve fi tting method was used by applying the least squares technique for non-linear models with respect to their parameters.

This study was conducted from March 16 to April 22, 2020, while Cuba was experiencing the impact of COVID-19, by a group of professors from the Mathematics Department at the Carlos Rafael Rodriguez University of Cienfuegos in collaboration with the Department of Educational Technology at the same institution.

Offi cial data on the number of confi rmed cases and deaths from COVID-19 reported by the governments of different countries were studied as summarized by WHO and recorded and published by Johns Hopkins University. These data are updated daily and show cumulative confi rmed cases, deaths and recoveries from the disease for different countries and territories. The fi rst record in this database is from January 22, 2020.[32] Data was collected until April 22, 2020.

For the countries studied, documentation began with the date of the fi rst recorded confi rmed cases or deaths in the territory (Table 1). The daily cumulative cases were recorded in both analyses. In Cuba, the fi rst cases were confi rmed on March 11, 2020, but they were recorded in the database the following day.

Table 1: First recorded date of confi rmed cases and deaths by country

First date recordedRecorded cumulative data

Cases DeathsItaly January 31, 2020 February 21, 2020Spain February 1, 2020 March 3, 2020Cuba March 12, 2020 March 18, 2020

Study variables The variables analyzed in this investigation are discrete quantitative variables, specifi cally:• Number of days elapsed since the fi rst positive cases of

COVID-19 were confi rmed. Each data point for this variable is recorded on a daily basis: for example, in the case of Cuba, the fi rst day corresponds to March 12, 2020 and the second cor-responds to March 13.

• Number of days elapsed since the fi rst confi rmed deaths of patients diagnosed with COVID-19. These values are recorded in a similar way to the previous variable, but using the database corresponding to deaths.

• Number of confi rmed daily cumulative cases for COVID-19.• Number of daily cumulative deaths for patients diagnosed with

COVID-19.

Data Management and Processing Downloaded daily as .csv fi les, data were decoded using programmed scripts for that purpose.

The Maxima 5.41.0[33] symbolic software programs and R 3.6.1[34] programming language for number processing were used to process the data.

To use the least squares method, the lsquare.mac (version 5.41.0) package was used in the Maxima program and for the commands for R; nls, SSlogis and SSgompertz from the stat package (version 3.6.1) and drm from the drc package (version 3.0-1) were used. To study the Root Mean Square Error (RMSE) and the signifi cance of the parameters of the model, the summary command from the stat package (version 3.6.1) was used and the adjusted R2 was calculated using rSquared from the miscTools package (version 0.6-22). To determine the goodness-of-fi t for the model, the command neill.test from the drc package (version 3.0-1) was used.

Analysis The logistic and Gompertz models were fitted to the data published for COVID-19 for confirmed cases and deaths in Spain and Italy. Italy had its peak of confirmed cases on March 26, 2020 and its peak deaths on March 27, 2020.[35] Spain had its peaks of confirmed cases and deaths on March 31, 2020 and April 2, 2020, respectively.[36] As of April 22, 2020, according to the Johns Hopkins database, Italy had reported a total of 187,327 confirmed cases due to COVID-19 with 25,085 deaths, while Spain had recorded 208,389 confirmed cases and 21,717 deaths. As these countries had passed the peak of the pandemic, the official published data on the peaks was compared to the forecasts obtained using the models.

The RMSE and the R2 adjusted coeffi cient of determination were calculated to study the goodness-of-fi t of the models, while keeping in mind that, for both models, values close to 1 for R2 and lower values for RMSE indicate a better fi t.

The models were adjusted to the data published for COVID-19 for confi rmed cases and deaths in Cuba. Goodness-of-fi t was determined using the analyses of R2 and RMSE. Signifi cance of the models’ adjusted coeffi cients was determined using the t test. Goodness-of-fi t was verifi ed using the Neill test, which is suitable for non-linear models with respect to the established parameters, and which utilizes grouping techniques in the event that there are no replicates.[37] The signifi cance threshold selected a priori was alpha = 0.05.

Once the models’ statistical significance had been demon-strated for distributions of confirmed COVID-19 cases and deaths in Cuba, these models were used to forecast the same.


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RESULTSConfi rmed cases of COVID-19Case Study, Italy The fi rst case was recorded on January 31, 2020. However, it was not until February 21 that exponential growth of the pandemic was offi cially reported. Figure 1 presents the geometric representation of cumulative confi rmed cases and the logistic model (Equation 3) and Gompertz curve (Equation 4). Table 2 presents the adjusted coeffi cients for each model, R2, the RMSE values obtained for each, and the forecasted peaks. Both models show an R2 greater than 0.99 with a notably lower RMSE in the Gompertz model. Using the logistic model, the peak was forecast at 60 days (March 30) after fi rst case, while the Gompertz model forecast it at 57 days (March 27).

Case Study, Spain The fi rst case was recorded on February 1, 2020. However, it was not until February 25 that the pandemic’s exponential growth was offi cially reported. Figure 2 shows the geometric representation of cumulative confi rmed cases, according to the logistic model (Equation 3) and Gompertz model (Equation 4).

Table 2 shows R2 greater than 0.99 for both models. The Gompertz model shows a lower RMSE than the logistic model, which suggests a better fi t. The estimated peak, according to the logistic model, is calculated at 62 days (April 2); while the estimated peak for the Gompertz model is estimated at 59 days (March 30).

Figure 1: Logistic model and Gompertz curve: confi rmed COVID-19 cases in Italy

Figure 2: Logistic model and Gompertz curve: confi rmed COVID-19 cases in Spain


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COVID-19 DEATHS Case Study, Italy The fi rst death was reported on Feb-ruary 21. The graph in Fig-ure 3 shows the geometric representation of observed cumulative deaths and the estimations by the logistic model (Equation 3) and Gompertz curve (Equation 4). Both models have an R2 greater than 0.99, however, the Gompertz model has a lower RMSE than the logistic model (Table 3). The logistic model has a forecasted peak at 41 days (April 1), while the

forecasted peak for the Gompertz model is 39 days (March 30) after the appearance of the fi rst case in the country (February 1).

Case Study, Spain The fi rst death was reported on March 3. Figure 4 shows the geometric representation of observed and predicted cases and deaths by the logistic model (Equation 3) and Gompertz model (Equation 4). Both models had an R2 higher than 0.99, however the Gompertz model had a smaller RMSE (Table 3). The logistic model had a projected peak at 33 days (April 4) while the projected peak for the Gompertz model is estimated at 30 days (April 1) after the reporting of the fi rst death in the country.

Estimation for Cuba The fi rst cases were diagnosed on March 11, recorded on March 12, and the fi rst death was on March 18. As of April 22, it had been 42 days since the fi rst report of infection and 36 days since the fi rst death. Figure 5 presents the geometric representation of observed cumulative confi rmed cases and deaths using the logistic model (Equation 3) and Gompertz curve model (Equation 4). On the graph, it can be observed that the models were correctly fi tted to the data and the increase in the data is within the prediction interval of 95%.

The model-generated forecasts for Cuba provide a projected peak of infection between 34 and 39 days after fi rst report of COVID-19 cases (March 12) and put the peak of deaths between 32 and 49 days after confi rmation of the fi rst death in the country (March 18). As with Spain and Italy, the Gompertz model forecast a greater total number of confi rmed cases and deaths than the logistic model. Table 4 shows the coeffi cients corresponding to the logistic models and Gompertz models fi tted to the reported Cuban data for confi rmed COVID-19 cases and deaths. The criteria for the goodness-of-fi t were similar for both models; they are slightly better in the Gompertz model for the distribution of confi rmed cases and in the logistic model for the distribution of deaths.

Associated p values for the signifi cance tests for the coeffi cients were all less than 0.05, indicating that the models were acceptable. Goodness-of-fi t was demonstrated using the Neill test, which presents levels of signifi cance higher than 0.05 for each model in each of the applied distributions (confi rmed cases and deaths). This also demonstrates an acceptable fi t for the models and thus their suitability for prognostic purposes.

Forecasts for the days with the highest numbers of infection and deaths were obtained using the calculation of the infl ection point

Table 2: Comparison of models fi t with real data for confi rmed cases of COVID-19 in Italy and SpainEstimate of confi rmed cases.

Coeffi cients(model) R2 RMSE Maximum cases

forecastedPeak

(forecasted) Real peak

Italy

Logistic ModelK = 186,701r = 0.13b = 7.54

>0.99 3537 186,701 (cumulative) March 30 March 26 (6203 cases

recorded)Gompertz ModelK = 216,522r = 0.07b = 44.97

>0.99 1291 216,522 (cumulative) March 27

Spain


>0.99 3644 204,499 (cumulative) April 2 March 31(9222 cases

recorded)Gompertz ModelK = 232,770r = 0.09b = 168.46

>0.99 1712 232,770 (cumulative) March 30

RMSE: Root Mean Square Error R2: coeffi cient of determination K: maximum number of individuals a population can sustain r: instantaneous rate of increase b: a measure of the displacement of the abscissa axis

Figure 3: Logistic model and Gompertz curve: COVID-19 deaths in Italy


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in each adjusted model and the cumulative totals corresponding to the K parameter (Table 4).

DISCUSSIONThe logistic growth and Gompertz models provided good forecasts for Italy and Spain. For both countries, the Gompertz model had better estimates for the peak in confi rmed cases and deaths. In the case of Italy, this model provided forecasts with an error of one day later and three days later for the peaks of infection and deaths respectively in comparison to the real peaks presented for that country. For Spain, the Gompertz model presented the forecasts for the peaks in infection and death with one day of error earlier than the real dates on which these peaks occurred. The Gompertz model forecast a higher total number of cases and deaths than the logistic model in both countries.

Figure 4: Logistic model and Gompertz curve: COVID-19 deaths in Spain Figure 5: Logistic model and Gompertz curve: confi rmed COVID-19 cases and deaths in Cuba


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The authors hypothesized that if the models provided good fore-casts for Spain and Italy, they would also do so for Cuba. Vari-ous authors[16–18,22–24] have used this subjective principle of plausibility and have anticipated goodness-of-fi t in territories

that had not yet passed the peak of the pandemic, based on adequate fi t in other territories that had passed their peaks.

To test this hypothesis, the models were fi tted to the dis-tribution of confi rmed cases and deaths recorded in Cuba and goodness-of-fi t was as-sessed. Signifi cance testing for the models’ coeffi cients demonstrated their validity. Each of the models passed the Neill goodness-of-fi t test, which makes it possible to generalize these models to mathematically describe the dynamics of the pandemic.

CONCLUSIONSThe logistic and Gompertz population growth models used to predict peaks and total numbers of infected cases and deaths due to COVID-19 have been sta-tistically validated with the usual analytical resources, which confi rmed the initial hypothesis that these mod-els could be extrapolated

and applied in Cuba. This provides two additional options that are methodologically viable to model epidemiological processes over time, especially for short-term forecasting and when the aim is not to include the infl uence of a large number of external factors.

Table 3: Comparison of models fi t with real data for deaths from COVID-19 in Italy and Spain

Estimate of deaths Coeffi cients(model) R2 RMSE Maximum cases

forecastedPeak

(forecasted) Real peak

Italy


>0.99 500.3 25,223 (cumulative) April 1 March 27(919

cases recorded)Gompertz Model

K = 29,826r = 0.07b = 15.11

>0.99 169.5 29,826 (cumulative) March 30

Spain


>0.99 454.5 21,494 (cumulative) April 4 April 2(961

cases recorded)Gompertz Model

K = 24,356r = 0.10b = 16.97

>0.99 137.3 24,356 (cumulative) April 1

RMSE: Root Mean Square Error K: maximum number of individuals a population can sustainr: instantaneous rate of increase b: a measure of the displacement of the abscissa axis

Table 4: Fitted models and their statistical signifi cance for reported data of confi rmed cases and deaths due to COVID-19 in Cuba. Forecasts of days peak of confi rmed cases and deaths and cumulative total

Models and forecasts for CubaConfi rmed cases Deaths

Logistic Model

Gompertz Model

Logistic Model

Gompertz Model

Model parameters and p-valueK = 1482 <0.001r = 0.15 <0.001b = 5.09 <0.001

K = 2678 <0.001r = 0.055 <0.001b = 8.27 <0.001

K = 63 <0.001r = 0.146 < 0.001b = 4.66 <0.001

K = 211 <0.001r = 0.039 <0.001b = 6.77 <0.001

Quality of fi t R2 = 0.999RMSE = 14.43

R2 = 0.999RMSE = 9.96

R2 = 0.997RMSE = 0.75

R2 = 0.996RMSE = 0.83

Neill goodness-of-fi t test F(p-value) 1.69 (0.199) 1.28 (0.290) 1.86 (0.189) 1.93 (0.174)

Forecast peak (day) April 14 April 19 April 18 May 5Total amount forecast 1482 2678 63 211RMSE: Root Mean Square Error K: maximum number of individuals a population can sustainr: instantaneous rate of increase b: a measure of the displacement of the abscissa axis

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THE AUTHORSJuan Felipe Medina-Mendieta (Correspond-ing author: [email protected]), computer sciences engineer with a master’s degree in new technologies. Assistant professor of math-ematics, Carlos Rafael Rodríguez University of Cienfuegos, Cuba. https://orcid.org/0000-0002-0508-9783

Manuel Cortés-Cortés, mathematician with a doctorate in economics. Full professor of math-ematics, Carlos Rafael Rodríguez University of Cienfuegos, Cuba. https://orcid.org/0000-0002-9903-3907

Manuel Cortés-Iglesias, computer sciences engineer with a master’s degree in applied mathematics. Assistant professor specializing in educational technology, Carlos Rafael Ro-dríguez University of Cienfuegos, Cuba. https://orcid.org/0000-0002-4517-9820

Submitted: April 29, 2020Approved for publication: July 17, 2020Disclosures: None

Published July 31, 2020 https:doi.org/10.37757/MR2020.V22.N3.8

COVID-19 Forecasts for Cuba Using Logistic Regression and Gompertz...

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